218 XIII INTEGRATION
(a) Show that the following properties are equivalent:
(a) The sequence (/xn) is convergent for the topology ^"e •
(ft) For each closed subset A of X, the sequence (/u,n(A)) has a finite limit.
(y) The sequence (fxn) converges for the topology ^"3 and satisfies the following
(C5) For each compact subset K of X and each e > 0, there exists an open
neighborhood U of K such that | jLtn|(U — K) g e for all n.
(8) The sequence (/xn) converges for the topology ^"3 and satisfies the following
(C6) For each e > 0 there exists 8 > 0 such that, for each universally measurable
set A satisfying v(A) ^ 8, we have |^i,|(A) < B for all n.
(To show that (/3) implies (y), use Problem 1 of Section 13.14 to show that the
sequence ||//,n|| is bounded. To establish (C5), argue by contradiction: first consider
the case in which the sequence (pn) tends to 0 for the topology ^"3 , and then pass to
the general case as in Problem l(d). Next show that (y) implies that the sequence
(/zn) is convergent for the topology &~s , and in particular implies (jS). To show that
(8) implies (a), use the definition of measurable functions. To show that (a) implies
(8), argue by contradiction. Finally, to show that (y) implies (a), consider first the case
where lim ||jun|| = ||ju,||, where ju is the limit of (fx,n) for the topology ^"5 , and argue by
contradiction to prove that (y) implies (8). To pass to the general case, reduce to the
situation where /z = 0, and argue by contradiction: we may assume that there exists
a function /e E6 such that the sequence (jU,n(/)) has a limit =£0. On the other hand, we
can pick a subsequence (finfc) of the sequence (/xn) such that the sequence (p.nk\ and
hence also (ftn~fc), converges vaguely. Hence arrive at a contradiction, by observing that
these two sequences also satisfy (Cs).)
(b) Give an example of a sequence (/xn) of measures belonging to M£(X, v) which
converges for the topology ^"4, v but not for the topology &"s (cf. Section 13.18, Problem
(c) Take X = [0, 1]. Show that the topologies &"$ and f6 on M£(X) are distinct
(cf. Section 12.15, Problem 2(c), and Section 13.11, Problem 3).
(d) Extend the results of Problems 1 and 2 to bounded complex measures.
3. Let X be a locally compact space, JJL a positive measure on X. Let (gn) be a sequence of
ju-integrable functions such that (1) the sequence (gn) converges in measure (Section
13.12, Problem 2) to a function g\ (2)' the sequence of measures (gn • IJL) converges for
the topology 3~$ (Problem 2). Show that under these conditions the function g is
/x-integrable and that the sequence (gn) tends to g in «^i(X, ft).
4. Let X be a compact space, p, a positive measure on X, and (/„) an orthonormal sequence
in ^c(^» fO consisting of functions which are uniformly bounded on X.
(a) Show that, for each function #e ^c<X /*)» tne sequence of numbers (#]/„) =
1 0(x)fn(x) dfji(x) tends to 0. (Reduce to the case where g is bounded and hence
belongs to :S?£(X, /*).)
(b) Let x0eX. For the series £(#|/,)/n(#o) to converge for every function
g e .5?£(X» /x), it is necessary and sufficient that (in the notation of Section 13.17,
Problem 2) the sequence of bounded measures Kn(jc0 , •) • ft should converge for the
topology «^"e (Problem 2) to a bounded measure hXQ - p,; hence in particular we have
/;(*o) - jhxo(x)fn(x) dtfx) for all n.pology «^~4, v (use (e)). Give an example of a